Ok, to wind back to what (OP says) actually happened:
When I picked him up it wasn’t obvious but Fil said “we nearly cut his hair but didn’t” I said I was glad they hadn’t. When we got home and he had a bath and hair wash it became obvious it had been cut.
DH is working away but he text her asking if she had cut it. No reply for a few days. Then a reply “I cut the tatty bits off, it’s hardly a short back and sides”
Be honest, how many of you would not be raging that someone actually dealt with you in this way when you pulled them up on something they'd done, which was your responsibility not theirs, which you'd asked them not to do and they'd gone and done it anyway??
OP said please don't do this with my child.
MIL does it. Seemingly lies to FIL about it, incidentally.
OP and her DH pull her up on it.
She takes days to have the manners to reply. Then she basically tells you quite rudely to swivel. 'It's hardly a short back and sides' OMG.
It's good that OP's next step is to double-check whether there was any acknowledgment that she was out of order. But from that description, I doubt it. Illustrates in spades what I outlined above - she feels that in some sense, your kids are her business too and she'll deal with things how she wants to.
Those folk saying let it go, grandparent relationship is more important etc. - it's because this relationship is important that this needs addressing. Send the message now that this isn't going to be the way things work, and with luck the message gets through and everyone knows the boundaries - as OhCalamity illustrates above. Leave it, and watch this happen with other things again and again, until you grow to dislike MIL and end up really not wanting her to be too close, or worse still finally falling out properly.
'No, we think nursery is best after all. The hair cutting thing really upset me and that's not fair on you, we don't want to fall out over doing things differently so it's really for the best that we separate grandparent time from childcare. Then you two can get to be granny and grandad not childcare! Much better for everyone :) '
She'll get the message. There's no relationship without trust, and you don't intend for her to think she gets to call the shots.